<s>
In	O
applied	O
mathematics	O
,	O
discretization	B-Algorithm
is	O
the	O
process	O
of	O
transferring	O
continuous	O
functions	O
,	O
models	O
,	O
variables	O
,	O
and	O
equations	O
into	O
discrete	O
counterparts	O
.	O
</s>
<s>
This	O
process	O
is	O
usually	O
carried	O
out	O
as	O
a	O
first	O
step	O
toward	O
making	O
them	O
suitable	O
for	O
numerical	B-General_Concept
evaluation	I-General_Concept
and	O
implementation	O
on	O
digital	O
computers	O
.	O
</s>
<s>
Dichotomization	B-Algorithm
is	O
the	O
special	O
case	O
of	O
discretization	B-Algorithm
in	O
which	O
the	O
number	O
of	O
discrete	O
classes	O
is	O
2	O
,	O
which	O
can	O
approximate	O
a	O
continuous	O
variable	O
as	O
a	O
binary	O
variable	O
(	O
creating	O
a	O
dichotomy	O
for	O
modeling	B-Application
purposes	O
,	O
as	O
in	O
binary	B-General_Concept
classification	I-General_Concept
)	O
.	O
</s>
<s>
Discretization	B-Algorithm
is	O
also	O
related	O
to	O
discrete	O
mathematics	O
,	O
and	O
is	O
an	O
important	O
component	O
of	O
granular	B-General_Concept
computing	I-General_Concept
.	O
</s>
<s>
In	O
this	O
context	O
,	O
discretization	B-Algorithm
may	O
also	O
refer	O
to	O
modification	O
of	O
variable	O
or	O
category	O
granularity	O
,	O
as	O
when	O
multiple	O
discrete	O
variables	O
are	O
aggregated	O
or	O
multiple	O
discrete	O
categories	O
fused	O
.	O
</s>
<s>
Whenever	O
continuous	O
data	O
is	O
discretized	B-Algorithm
,	O
there	O
is	O
always	O
some	O
amount	O
of	O
discretization	B-Algorithm
error	I-Algorithm
.	O
</s>
<s>
The	O
goal	O
is	O
to	O
reduce	O
the	O
amount	O
to	O
a	O
level	O
considered	O
negligible	O
for	O
the	O
modeling	B-Application
purposes	O
at	O
hand	O
.	O
</s>
<s>
The	O
terms	O
discretization	B-Algorithm
and	O
quantization	B-Algorithm
often	O
have	O
the	O
same	O
denotation	O
but	O
not	O
always	O
identical	O
connotations	O
.	O
</s>
<s>
The	O
same	O
is	O
true	O
of	O
discretization	B-Algorithm
error	I-Algorithm
and	O
quantization	B-Algorithm
error	O
.	O
</s>
<s>
Mathematical	O
methods	O
relating	O
to	O
discretization	B-Algorithm
include	O
the	O
Euler	O
–	O
Maruyama	O
method	O
and	O
the	O
zero-order	O
hold	O
.	O
</s>
<s>
Discretization	B-Algorithm
is	O
also	O
concerned	O
with	O
the	O
transformation	O
of	O
continuous	O
differential	O
equations	O
into	O
discrete	O
difference	O
equations	O
,	O
suitable	O
for	O
numerical	B-General_Concept
computing	I-General_Concept
.	O
</s>
<s>
The	O
equation	O
for	O
the	O
discretized	B-Algorithm
measurement	O
noise	O
is	O
a	O
consequence	O
of	O
the	O
continuous	O
measurement	O
noise	O
being	O
defined	O
with	O
a	O
power	O
spectral	O
density	O
.	O
</s>
<s>
Where	O
and	O
are	O
the	O
discretized	B-Algorithm
state-space	O
matrices	O
.	O
</s>
<s>
Numerical	B-General_Concept
evaluation	I-General_Concept
of	O
is	O
a	O
bit	O
trickier	O
due	O
to	O
the	O
matrix	O
exponential	O
integral	O
.	O
</s>
<s>
The	O
discretized	B-Algorithm
process	O
noise	O
is	O
then	O
evaluated	O
by	O
multiplying	O
the	O
transpose	O
of	O
the	O
lower-right	O
partition	O
of	O
G	O
with	O
the	O
upper-right	O
partition	O
of	O
G	O
:	O
</s>
<s>
which	O
is	O
an	O
exact	O
solution	O
to	O
the	O
discretization	B-Algorithm
problem	O
.	O
</s>
<s>
Exact	O
discretization	B-Algorithm
may	O
sometimes	O
be	O
intractable	O
due	O
to	O
the	O
heavy	O
matrix	O
exponential	O
and	O
integral	O
operations	O
involved	O
.	O
</s>
<s>
This	O
is	O
also	O
known	O
as	O
the	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
,	O
which	O
is	O
also	O
known	O
as	O
the	O
forward	B-Algorithm
Euler	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
Other	O
possible	O
approximations	O
are	O
,	O
otherwise	O
known	O
as	O
the	O
backward	O
Euler	B-Algorithm
method	I-Algorithm
and	O
,	O
which	O
is	O
known	O
as	O
the	O
bilinear	O
transform	O
,	O
or	O
Tustin	O
transform	O
.	O
</s>
<s>
In	O
statistics	O
and	O
machine	O
learning	O
,	O
discretization	B-Algorithm
refers	O
to	O
the	O
process	O
of	O
converting	O
continuous	O
features	O
or	O
variables	O
to	O
discretized	B-Algorithm
or	O
nominal	O
features	O
.	O
</s>
<s>
and	O
is	O
the	O
(	O
unitary	O
,	O
ordinary	O
frequency	O
)	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
Functions	O
which	O
are	O
not	O
smooth	O
can	O
be	O
made	O
smooth	O
using	O
a	O
mollifier	B-Algorithm
prior	O
to	O
discretization	B-Algorithm
.	O
</s>
<s>
As	O
an	O
example	O
,	O
discretization	B-Algorithm
of	O
the	O
function	O
that	O
is	O
constantly	O
yields	O
the	O
sequence	O
which	O
,	O
interpreted	O
as	O
the	O
coefficients	O
of	O
a	O
linear	O
combination	O
of	O
Dirac	O
delta	O
functions	O
,	O
forms	O
a	O
Dirac	O
comb	O
.	O
</s>
<s>
If	O
additionally	O
truncation	B-Algorithm
is	O
applied	O
,	O
one	O
obtains	O
finite	O
sequences	O
,	O
e.g.	O
</s>
